Part 1 — THE POLYTECHNIC SCHOOL OF FRANCE. (continued 2)

The engineers of the Ponts et Chaussées prepare the projects and plans, and direct the execution of the works for the construction, preservation, and repair of high roads, and of the bridges and other structures connected with these roads, with navigable rivers, canals, seaports, lighthouses, &c. They are charged with the superintendence of railways, of works for draining marshes, and operations affecting water-courses; they report upon applications to erect factories driven by water. Under certain circumstances, they share with the Mining Engineers the duty of inspecting steam-engines.

Permission is not unfrequently granted to the engineers of the Ponts et Chaussées to accept private employment. They receive leave of absence for a certain time, retaining their rank and place in their corps, but without pay.

Mining Engineers.—(Mines.)

The Mining School of Application is organized almost exactly on the same plan as that of the Ponts et Chaussées: like the latter, it is in Paris.

The course of instruction, which lasts three years, consists of lectures, drawing, chemical manipulation and analysis, visits to manufactories, geological excursions, and the preparation of projects for mines and machines. Journeys are made by the pupils, during the second half of the last two years of the course, into the mineral districts of France or foreign countries for the purpose of studying the practical details of mining. These journeys last one hundred days at least. The pupils are required to examine carefully the railroads and the geological features of the countries they pass through, and to keep a journal of facts and observations. In the final examination, marks are given for every part of their work.

The mining engineers, when stationed in the departments, are charged to see that the laws and ordinances relating to mines, quarries, and factories are properly observed, and to encourage, either directly or by their advice, the extension of all branches of industry connected with the extraction and treatment of minerals.

One of their principal duties is the superintendence of mines and quarries, in the three-fold regard of safety of the workmen, preservation of the soil, and economical extraction of the minerals.

They exercise a special control over all machines designed for the production of steam, and over railways, as far as regards the metal and fuel.

The instructors in the School of Application in Paris, and in the School of Mines at St. Etienne, are exclusively taken from the members of the corps.

Like the engineers of the Ponts et Chaussées, the mining engineers obtain permission to undertake private employment.

Tobacco Department.—(Administration des Tabacs.)

The pupils who enter the tobacco service, commence, on quitting the Polytechnic, with the rank of élève de 2^e classe. They study, in the manufactory at Paris, chemistry, physics, and mechanics, as applied to the preparation of tobacco. They make themselves acquainted at the same time with the details of the manufacture and with the accounts and correspondence.

They are generally promoted to the rank of élevè de 1^re classe in two years. They rise afterwards successively to the rank of sub-inspector, inspector, and director.

After completing their instruction at the manufactory of Paris, the élevès are sent to tobacco manufactories in other parts of France.

Promotion in the tobacco service does not follow altogether by seniority. Knowledge of the manufacture and attention to their duties are much considered, as the interests of the treasury are involved in the good management of the service.

Telegraphs.—(Lignes Telégraphiques.)

On entering the telegraphic service the pupils of the Polytechnic receive the rank of élevè inspecteur.

They pass the first year at the central office. During the six winter months they study, under two professors, the composition of signals, and the regulations which insure their correctness and dispatch, the working of telegraphs and the manner of repairing them, the theory of the mode of tracing lines and of determining the height of the towers, electro-magnetism and its application to the electric telegraph. During the summer months they make tours of inspection. They assist in the execution of works, and practice leveling and the laying down of lines.

At the end of the year the élevès inspecteurs undergo an examination, and, if there are vacancies, are appointed provisional inspectors. After a year in this rank they may be appointed inspectors either in France or Algeria.

Each inspector has charge of a district containing from twelve to fifteen stations. He is obliged to make a tour of inspection once a month of at least ten days’ duration.

After a certain number of years’ service the inspector rises to the rank of director. Besides their other duties, the directors exercise a general superintendence over the inspectors.

PROGRAMMES OF THE PRINCIPAL COURSES OF INSTRUCTION OF THE IMPERIAL POLYTECHNIC SCHOOL DURING THE TWO YEARS OF STUDY.

I. ANALYSIS.—FIRST YEAR.

DIFFERENTIAL CALCULUS.

Lessons 1–9. Derivatives and Differentials of Functions of a Single Variable.

Indication of the original problems which led geometers to the discovery of the infinitesimal calculus.

Use of infinitesimals; condition, subject to which, two infinitely small quantities may be substituted for one another. Indication in simple cases of the advantage of such substitution.

On the different orders of infinitely small quantities. Infinitely small quantities of a certain order may be neglected in respect of those of an inferior order. The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit.

Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given.

A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. Concavity and convexity of curves; points of inflection.

Principle of function of functions. Differentiation of inverse functions.

Differentials of the sums, products, quotients, and powers of functions, whose differentials are known. General theorem for the differentiation of functions composed of several functions.

Differentials of exponential and logarithmic functions.

Differentials of direct and inverse circular functions.

Differentiation of implicit functions.

Tangents to curves of double curvature. Normal plane.

Differential of the area and arc of a plane curve, in terms of rectilinear and polar co-ordinates.

Differential of the arc of a curve of double curvature.

Applications to the cycloid, the spiral of Archimedes, the logarithmic spiral, the curve whose normal, sub-normal, or tangent, is constant; the curve whose normal passes through a fixed point; the curve whose arc is proportional to the angle which it subtends at a given point.

Derivatives and differentials of different orders of functions of one variable. Notation adopted.

Remarks upon the singular points of plane curves.

Lessons 10–13. Derivatives and Differentials of Functions of Several Variables.

Partial derivatives and differentials of functions of several variables. The order in which two or any number of differentiations is effected does not influence the result.

Total differentials. Symbolical formula for representing the total differential of the n^th order of a function of several independent variables.

Total differentials of different orders of a function; several dependent variables. Case where these variables are linear functions of the independent variables.

The infinitesimal increment of a function of several variables may in general be regarded as a linear function of the increments assigned to the variables. Exceptional cases.

Tangent and normal planes to curved surfaces.

Lessons 14–18. Analytical Applications of the Differential Calculus.

Development of F(x + h,) according to ascending powers of h. Limits within which the remainder is confined on stopping at any assigned power of h.

Development of F(x,) according to powers of x or x - a; a being a quantity arbitrarily assumed. Application to the functions sin(x,) cos x, a^x, (1 + x^m) and log.(1 + x.) Numerical applications. Representation of cos x and sin x by imaginary exponential quantities.

Developments of cos^m x and sin^m x in terms of sines and curves of multiples of x.

Development of F(x + h, y + k,) according to powers of h and k. Development of F(x, y) according to powers of x and y. Expression for the remainder. Theorem on homogeneous functions.

Maxima and minima of functions of a single variable; of functions of several variables, whether independent or connected by given equations. How to discriminate between maxima and minima values in the case of one and two independent variables.

True values of functions, which upon a particular supposition assume one or another of the forms

00, ∞∞, ∞ + 0, 0⁰, 4^∞

Lessons 19–23. Geometrical Applications. Curvature of Plane Curves.

Definition of the curvature of a plane curve at any point. Circle of curvature. Center of curvature. This center is the point where two infinitely near normals meet.

Radius of curvature with rectilinear and polar co-ordinates. Change of the independent variable.

Contacts of different orders of plane curves. Osculating curves of a given kind. Osculating straight line. Osculating circle. It is identical with the circle of curvature.

Application of the method of infinitesimals to the determination of the radius of curvature of certain curves geometrically defined. Ellipse, cycloid, epicycloid, &c.

Evolutes of plane curves. Value of the arc of the evolute. Equation to the involute of a curve. Application to the circle. Evolutes considered as envelops. On envelops in general. Application to caustics.

Lessons 24–27. Geometrical Applications continued. Curvature of Lines of Double Curvature and of Surfaces.

Osculating plane of a curve of double curvature. It may be considered as passing through three points infinitely near to one another, or as drawn through a tangent parallel to the tangent infinitely near to the former. Center and radius of curvature of a curve of double curvature. Osculating circle. Application to the helix.

Radii of curvature of normal s of a surface. Maximum and minimum radii. Relations between these and that of any , normal or oblique.

Use of the indicatrix for the demonstration of the preceding results. Conjugate tangents. Definition of the lines of curvature. Lines of curvature of certain simple surfaces. Surface of revolution. Developable surfaces. Differential equation of lines of curvature in general.

Lesson 28. Cylindrical, Conical, Conoidal surfaces, and Surfaces of Revolution.

Equations of these surfaces in finite terms. Differential equations of the same deduced from their characteristic geometrical properties.

INTEGRAL CALCULUS.

Lessons 29–34. Integration of Functions of a Single Variable.

Object of the integral calculus. There always exists a function which has a given function for its derivative.

Indefinite integrals. Definite integrals. Notation. Integration by separation, by substitution, by parts.

Integration of rational differentials, integer or fractional, in the several cases which may present themselves. Integration of the algebraical differentials, which contain a radical of the second degree of the form √c+bx+ax². Different transformations which render the differential rational. Reduction of the radical to one of the forms

√x²+x², √a²-x², √x²-a².

Integration of the algebraical differentials which contain two radicals of the form

√a+x, √b+x,

or any number of monomials affected with fractional indices. Application to the expressions

x^mdx dx x^mdx

√1-x² , x^m √1-x² , √ax-x

Integration of the differentials

F(log x) dx x , F sin^-1x dx √1-x² , x(log xⁿ)dx, x^m e^ax dx, (sin^-1x^m)dx.

Integration of the differentials e^ax sin bxdx and e^ax cos bxdx.

Integration of (sin x^m.)(cos xⁿ) dx.

Integration by series. Application to the expression

dx

√ax-x² √1-bx

Application of integration by series to the development of functions, the development of whose derivatives is given: tan^-1x, sin^-1x, log(1 + x.)

Lessons 35–38. Geometrical Applications.

Quadrature of certain curves. Circle, hyperbola, cycloid, logarithmic spiral, &c.

Rectification of curves by rectilinear or polar co-ordinates. Examples. Numerical applications.

Cubic content of solids of revolution. Quadrature of their surfaces.

Cubic content of solids in general, with rectilinear or polar co-ordinates. Numerical applications.

Quadrature of any curved surfaces expressed by rectangular co-ordinates. Application to the sphere.

Lessons 39–42. Mechanical Applications.

General formula for the determination of the center of gravity of solids, curved or plane surfaces, and arcs of curves. Various applications.

Guldin’s theorem.

Volume of the truncated cylinder.

General formula which represent the components of the attraction of a body upon a material point, upon the supposition that the action upon each element varies inversely as the square of the distance. Attraction of a spherical shell on an external or internal point.

Definition of moments of inertia. How to calculate the moment of inertia of a body in relation to a straight line, when the moment in relation to a parallel straight line is known. How to represent the moments of inertia of a body relative to the straight lines which pass through a given point by means of the radii vectores of an ellipsoid. What is meant by the principal axes of inertia.

Determination of the principal moments of inertia of certain homogeneous bodies, sphere, ellipsoid, prism, &c.

Lessons 43–45. Calculus of Differences.

Calculation of differences of different orders of a function of one variable by means of values of the function corresponding to equidistant values of the variable.

Expression for any one of the values of the function by means of the first, and its differences. Numerical applications; construction of tables representing a function whose differences beyond a certain order may be neglected. Application to the theory of interpolation. Formulæ for approximation by quadratures. Numerical exercises relative to the area of equilateral hyperbola or the calculation of a logarithm.

Lessons 46–48. Revision.

General reflections on the subjects contained in the preceding course.

ANALYSIS.—SECOND YEAR.

CONTINUATION OF THE INTEGRAL CALCULUS.

Lessons 1–2. Definite Integrals.

Differentiation of a definite integral with respect to a parameter in it, which is made to vary. Geometrical demonstration of the formula. Integration under the sign of integration. Application to the determination of certain definite integrals.

Determination of the integrals ∫ sin ax x dx, and ∫ cos bx sin ax x dx, between the limits 0 and x. Remarkable discontinuity which these integrals present.

Determination of ∫e^-x²dx and ∫e^-x²cos mx dx between the limits 0 and ∞.

Lesson 3. Integration of Differentials containing several Variables.

Condition that an expression of the form M dx + N dy in which M and N are given functions of x and y may be an exact differential of two independent variables x and y. When this condition is satisfied, to find the function.

Extension of this theory to the case of three variables.

Lessons 4–6. Integration of Differential Equations of the First Order.

Differential equations of the first order with two variables. Problem in geometry to which these equations correspond. What is meant by their integral. This integral always exists, and its expression contains an arbitrary constant.

Integration of the equation M dx + N dy = 0 when its first member is an exact differential. Whatever the functions M and N may be there always exists a factor µ, such that µ (M dx + N dy) is an exact differential.

Integration of homogeneous equations. Their general integral represents a system of similar curves. The equation (a + b x + c y) dx + (a’ + b’ x + c’ y) dy = c, may be rendered homogeneous. Particular case where the method fails. How the integration may be effected in such case.

Integration of the linear equation of the first order dy dx + P y = Q, where P and Q denote functions of x. Examples.

Remarks on the integration of equations of the first order which contain a higher power than the first of dy dx . Case in which it may be resolved in respect of dy dx . Case in which it may be resolved in respect of x or y.

Integrations of the equation y = x dy dx + φ ( dy dx ). Its general integral represents a system of straight lines. A particular solution represents the envelop of this system.

Solution of various problems in geometry which lead to differential equations of the first order.

Lessons 7–8. Integration of Differential Equations of Orders superior to the First.

The general integral of an equation of the m order contains m arbitrary constants.

(The demonstration is made to depend on the consideration of infinitely small quantities.)

Integration of the equation d^my dx^m = φ (x.)

Integration of the equation d²y dx² = φ (y, dy dx ).

How this is reduced to an equation of the first order. Solution of various problems in geometry which conduct to differential equations of the second order.

Lessons 9–10. On Linear Equations.

When a linear equation of the m^th order contains no term independent of the unknown function and its derivatives, the sum of any number whatever of particular integrals multiplied by arbitrary constants is also an integral. From this the conclusion is drawn that the general integral of this equation is deducible from the knowledge of m particular integrals.

Application to linear equations with constant co-efficients. Their integration is made to depend on the resolution of an algebraical equation. Case where this equation has imaginary roots. Case where it has equal roots. The general integral of a linear equation of any order, which contains a term independent of the function, may be reduced by the aid of quadratures to the integration of the same equation with this term omitted.

Lesson 11. Simultaneous Equations.

General considerations on the integration of simultaneous equations. It may be made to depend on the integrations of a single differential equation. Integration of a system of two simultaneous linear equations of the first order.

Lesson 12. Integrations of Equations by Series.

Development of the unknown function of the variable x according to the powers of x-a. In certain cases only a particular integral is obtained. If the equation is linear, the general integral may be deduced from it by the variation of constants.

Lessons 13–16. Partial Differential Equations.

Elimination of the arbitrary functions which enter into an equation by means of partial derivatives. Integration of an equation of partial differences with two independent variables, in the case where it is linear in respect to the derivatives of the unknown function. The general integral contains an arbitrary function.

Indication of the geometrical problem, of which the partial differential equation expresses analytically the enunciation. Integration of the partial differential equations to cylindrical, conical, conoidal surfaces of revolution. Determination of the arbitrary functions.

Integration of the equation d²u dy² = a² d²u dx². The general integral contains two arbitrary functions. Determination of these functions.

Lessons 17–23. Applications to Mechanics.

Equation to the catenary.

Vertical motion of a heavy particle, taking into account the variation of gravity according to the distance from the center of the earth. Vertical motion of a heavy point in a resisting medium, the resistance being supposed proportional to the square of the velocity.

Motion of a heavy point compelled to remain in a circle or cycloid. Simple pendulum. Indication of the analytical problem to which we are led in investigating the motion of a free point.

Motion of projectiles in a vacuum. Calculation of the longitudinal and transversal vibrations of cords. Longitudinal vibrations of elastic rods. Vibration of gases in cylindical tubes.

Lessons 24–26. Applications to Astronomy.

Calculation of the force which attracts the planets, deduced from Kepler’s laws. Numerical data of the question.

Calculation of the relative motion of two points attracting one another, according to the inverse square of the distance.

Determination of the masses of the earth and of the planets accompanied by satellites. Numerical applications.

Lessons 27–30.

Elements of the calculus of probabilities and social arithmetic.

General principles of the calculus of chances. Simple probability, compound probability, partial probability, total probability. Repeated trials. Enunciation of Bernouilli’s theorem (without proof.)

Mathematical expectation. Applications to various cases, and especially to lotteries.

Tables of population and mortality. Mean life annuities, life interests, assurances, &c.

Lessons 31–32. Revision.

General reflections on the subjects comprised in the course.

II. DESCRIPTIVE GEOMETRY AND STEREOTOMY.

General Arrangements.

The pupils take in the lecture-room notes and sketches upon sheets, which are presented to the professor and the “répétiteurs” at each interrogation. The care with which these notes are taken is determined by “marks,” of which account is taken in arranging the pupils in order of merit.

The plans are made according to programmes, of which the conditions are different for different pupils. The drawings are in general accompanied with decimal scales, expressing a simple ratio to the meter. They carry inscriptions written conformably to the admitted models, and are, when necessary, accompanied with verbal descriptions.

In the graphic exercises of the first part of the course, the principal object is to familiarize the pupils with the different kinds of geometrical drawing, such as elevations and shaded s, oblique projections and various kinds of perspective. The pupils are also accustomed to different constructions useful in stereotomy.

The subjects for graphic exercises in stereotomy are taken from roofs, vaults, and staircases. Skew and oblique arches are the subject of detailed plans.

FIRST YEAR.

DESCRIPTIVE GEOMETRY.—GEOMETRICAL DRAWING.

Lessons 1–3. Revision and Completion of the Subjects of Descriptive Geometry comprised in the Programme for Admission into the School.

Object of geometrical drawing. Methods of projection. Representation of points, lines, planes, cones, cylinders, and surfaces of revolution. Construction of tangent planes to surfaces, of curves, of inter of surfaces, of their tangents and their assymplotes.

Osculating plane of a curve of double curvature. A curve in general cuts its osculating plane.

When the generating line of a cylinder or a cone becomes a tangent to the directrix, the cylinder or cone in general has an edge of regression along this generating line. The osculating plane of the directrix at the point of contact touches the surface along this edge.

Projections of curves of double curvature; infinite branches and their assymplotes, inflections, nodes, cusps, &c.

Change of planes of projection.

Reduction of scale; transposition.

Advantage and employment of curves of error; their irrelevant solutions.

Lessons 4–6. Modes of Representation for the Complete Definition of Objects.

Representation by plans, s, and elevation.

Projection by the method of contours. Representation of a point, a line, and a plane; questions relative to the straight line and plane. Representation of cones and cylinders; tangent planes to these surfaces.

Lessons 7–11. Modes of Representation which are not enough in themselves to define objects completely.

Isometrical and other kinds of perspective.

Oblique projections.

Conical perspective: vanishing points; scales of perspective; method of squares; perspective of curved lines; diverse applications. Choice of the point of sight. Rules for putting an elevation in perspective. Rule for determining the point of sight of a given picture, and for passing from the perspective to the plan as far as that is possible. Perspective of reflected images. Notions on panoramas.

Lessons 12–13. Representations with Shadows.

General observations on envelops and characteristics.

A developable surface is the envelop of the position of a movable plane; it is composed of two sheets which meet. It may be considered as generated by a straight line, which moves so as to remain always a tangent to a fixed curve.

Theory of shade and shadow, of the penumbra, of the brilliant point, of curves of equal intensity, of bright and dark edges.

Atmospheric light: direction of the principal atmospheric ray. Notions on the degradation of tints; construction of curves of equal tint.

Influence of light reflected by neighboring bodies.

Received convention in geometrical drawing on the direction of the luminous ray, &c.

Perspective of shadows.

Lessons 14–15. Construction of Lines of Shadows and of Perspective of Surfaces.

Use of circumscribed cones and cylinders, and of the normal parallel to a given straight line.

General method of construction of lines of shadow and of perspective of surfaces by plane s and auxiliary cylindrical or conical surfaces.

Construction of lines of shadow and perspective of a surface of revolution.

The curve of contact of a cone circumscribed about a surface of the second degree is a plane curve. Its plane is parallel to the diametral plane, conjugate to the diameter passing through the summit of the cone. The curve of contact of a cylinder circumscribed about a surface of the second degree is a plane curve, and situated in the diametral plane conjugate to the diameter parallel to the axis of the cylinder.

The plane parallel s of a surface of the second degree are similar curves. The locus of their centers is the diameter conjugate to that one of the secant planes which passes through the center of the surface.

General study of surfaces with reference to the geometrical constructions to which their use gives rise.

Lesson 16. Complementary Notions on Developable Surfaces.

Development of a developable surface; construction of transformed curves and their tangents. Developable surface; an envelop of the osculating planes of a curve. The osculating plane of a curve at a given point may be constructed by considering it as the edge of regression of a developable surface; this construction presents some uncertainty in practice. Notions on the helix and the developable helicoid.

Approximate development of a segment of an undevelopable surface.

Lessons 17–18. Hyperbolic Paraboloid.

Double mode of generation of the paraboloid by straight lines; plane-directers; tangent planes, vertex, axis, principal planes; representation of this surface. Construction of the tangent plane parallel to a given plane. Construction of plane s and of curves of contact, of cones, and circumscribed cylinders.

Scalene paraboloid. Isosceles paraboloid.

Identity of the paraboloid with one of the five surfaces of the second degree studied in analytical geometry.

Re-statement without demonstration of the properties of this surface found by analysis, principally as regards its generation by the conic s.

Lessons 19–20. General Properties of Warped or Ruled Surfaces.

Principal modes of generation of warped surfaces. When two warped surfaces touch in three points of a common generatrix, they touch each other in every point of this straight line. Every plane passing through a generatrix touches the surface at one point in this line. The tangent plane at infinity is the plane-directer to all the paraboloids of “raccordement.”

Construction of the tangent planes and curves of contact of circumscribed cones and cylinders. When two infinitely near generatrices of a warped surface are in the same plane, all the curves of contact of the circumscribed cones and cylinders pass through their point of concourse.

The normals to a warped surface along a generatrix form an isosceles paraboloid. The name of central point of a generatrix is given to the point where it is met by the straight line upon which is measured its shortest distance from the adjoining generatrix. The locus of these points forms the line of striction of the surface. The vertex of the normal paraboloid along a generating line is situated at the central point. If the point of contact of a plane touching a warped surface moves along a generatrix, beginning from the central point, the tangent of the angle which the tangent plane makes with its primitive position is proportional to the length described by the point of contact. The tangent plane at the central point is perpendicular to the tangent plane at infinity upon the same generatrix. Construction of the line of striction by aid of this property.

Lessons 21–22. Ruled Surfaces with plane-divecters Conoids.

The plane-directer of the surface is also so to all the paraboloids of “raccordement.” Construction of the tangent planes and curves of contact of the circumscribed cones and cylinders.

The line of striction of the surface is its curve of contact with a circumscribed cylinder perpendicular to the directer-plane. Determination of the nature of the plane s.

The lines of striction of the scalene paraboloid are parabolas; those of the isosceles paraboloid are straight lines.

Construction of the tangent plane parallel to a given plane.

Conoid: discussion of the curves of contact of the circumscribed cones and cylinders.

Right conoid. Conoid whose inter with a torus of the same height, whose axis is its rectilinear directrix, has for its projection upon the directer-plane two arcs of Archimedes’ spiral. Construction of the tangents to this curve of inter.

Lessons 23–25. Ruled Surfaces which have not a Directer-Plane. Hyperboloid. Surface of the “biais passe.”

Directer-cone: its advantages for constructing the tangent plane parallel to a given plane, and for determining the nature of the plane s. The tangent planes to the points of the surface, situated at infinity, are respectively parallel to the tangent plane of the directer-cone. Developable surface which is the envelope of these tangent planes at infinity. Construction of a paraboloid of raccordement to a ruled surface defined by two directrices and a directrix cone.

Hyperboloid; double mode of generation by straight lines; center; assymptotic cone.

Scalene hyperboloid; hyperboloid of revolution. Identity of the hyperboloid with one of the five surfaces of the second degree studied in analytical geometry.

Re-statement without demonstration of the properties of this surface, found by analysis, principally as to what regards the axis, the vertices, the principal planes, and the generation by conic s.

Hyperboloid of raccordement to a ruled surface along a generatrix; all their centers are in the same plane. Transformation of a hyperboloid of raccordement.

Surface of the biais passé. Construction of a hyperboloid of raccordement; its transformation into a paraboloid.

Construction of the tangent plane at a given point.

Lessons 26–28. Curvature of Surfaces. Lines of Curvature.

Re-statement without proof of the formula of Euler given in the course of analysis.

There exists an infinity of surfaces of the second degree, which at one of their vertices osculate any surface whatever at a given point.

In the tangent plane, at a point of a surface, there exists a conic , whose diameters are proportional to the square roots of the radii of curvature of the normal s to which they are tangents. This curve is called the indicatrix. It is defined in form and position, but not in magnitude. The normal s tangential to the axes of the indicatrix are called the principal s.

The indicatrix an ellipse; convex surfaces; umbilici; line of spherical curvatures.

The indicatrix a hyperbola; surfaces with opposite curvatures.

The assymplotes of the indicatrix have a contact of the second order with the surface, and of the first order with the of the surface by its tangent plane.

A ruled surface has contrary curvatures at every point. The second assymplotes of the indicatrices of all the points of the same generatrix form a hyperboloid, if the surface has not directer-plane,—a paraboloid, if it have one.

Curvature of developable surfaces.

There exists upon every surface two systems of orthogonal lines, such that every straight line subject to move by gliding over either of them, and remaining normal to the surface, will engender a developable surface. These lines are called lines of curvature.

The two lines of curvature which cross at a point, are tangents to the principal s of the surface at that point.

Remarks upon the lines of curvature of developable surfaces, and surfaces of revolution.

Determination of the radii of curvature, and assymplotes of the indicatrix at a point of a surface of revolution.

Lessons 29–30. Division of Curves of Apparent Contour, and of Separation of Light and Shadow into Real and Virtual Parts.

When a cone is circumscribed about a surface, at any point whatever of the curve of contact, the tangent to this curve and the generatrix of the cone are parallel to two conjugate diameters of the indicatrix.

Surfaces, as they are considered in shadows, envelop opaque bodies, and the curve of contact of a circumscribed cone, only forms a separation of light and shadow, for a luminous point at the summit of the cone, when the generatrices of this cone are exterior. This line is thus sometimes real and sometimes virtual.

Upon a convex surface, the curve of separation of light and shade is either all real or all virtual. Upon a surface with contrary curvatures, this curve presents generally a succession of real and virtual parts: the curve of shadow cast from the surface upon itself presents a like succession. These curves meet tangentially, and the transition from the real to the virtual parts upon one and the other, take place at their points of contact in such a way that the real part of the curve of shadow continues the real part of the curve of separation of light and shade. The circumscribed cones have edges of regression along the generatrices, which correspond to the points of transition.

The lines of visible contour present analogous circumstances.

General method of determining the position of the transition points. Special method for a surface of revolution.

Lessons 31–34. Ruled Helicoidal Surfaces.

Surface of the thread of the triangular screw; generation, representation, s by planes and conical cylinders.

Construction of the tangent plane at a given point, or parallel to a given plane. The axis is the line of striction.

Construction of lines of shadow and perspective: their infinite branches, their assymplotes. Determination of the osculating hyperboloid along a generatrix.

Representation and shading of the screw with a triangular thread and its nut.

Surface of the thread of the square screw; generation, s by planes and conical cylinders; tangent planes; curve of contact of a circumscribed cone.

The curve of contact of a circumscribed cylinder is a helix whose step is half that of the surface. Determination of the osculating paraboloid. At any point whatever of the surface, the absolute lengths of the radii of curvature are equal.

Representation and shading of the screw with a square thread, and of its nut.

Observations on the general ruled helicoidal surface, and on the surface of intrados of the winding staircase.

Lesson 35. Different Helicoidal Surfaces.

Saint-Giles screw, worm-shaped screw and helicoidal surfaces to any generatrix. Every tangent to the meridian generatrix describes a screw surface with triangular thread, which is circumscribed about the surface, along a helix, and may be used to resolve the problems of tangent planes, circumscribed cylinders, &c.

Helicoid of the open screw, its generation, tangent planes.

Lessons 36–37. Topographical Surfaces.

Approximate representation of a surface by the figured horizontal projections of a series of equidistant horizontal s. This method of representation is especially adapted to topographical surfaces, that is to say, surfaces which a vertical line can only meet in one point.

Lines of greatest slope. Trace of a line of equal slope between two given points.

Inter of a plane and a surface, of two surfaces, of a straight line and a surface.

Tangent planes, cones, and cylinders circumscribed about topographical surfaces.

Use of a topographical surface to replace a table of double-entry when the function of two variables, which it represents, is continuous. It is often possible, by a suitable anamorphosis, to make an advantageous transformation in the curves of level.

Lesson 38. Revision.

Review of the different methods of geometrical drawing. Advantages and disadvantages of each.

Comparison of the different kinds of surfaces, résumé of their general properties.

Object, method, and spirit of descriptive geometry.

SECOND YEAR.

STEREOTOMY.—WOOD-WORK.

Lessons 1–4. Generalities.

Notions on the mode of action of forces in carpentry. Resistance of a piece of wood to a longitudinal effort and to a transversal effort. Distinction between resistance to flexure and resistance to rupture. Beams.

Advantages of the triangular system, St. Andrew’s cross.

Lessons 5–8. Roofs.

Ordinary composition of roofs.

Distribution of pressures in the different parts of a girded roof.

Design of the different parts of roofs, &c., &c.

Lessons 9–10. Staircases.

MASONRY.

Lessons 11–12. Generalities.

Notions on the settlement of vaulted roofs. Principal forms of vaults, en berceau, &c., &c.

Distribution of the pressures, &c.

Division of the intrados. Nature of the surfaces at the joints, &c., &c.

Lessons 13–15. Berceaux and descentes.

Lessons 16–22. Skew Arches.

Study of the general problem of skew arches.

First solution. Straight arches en échelon.

Second solution: Orthogonal appareil. True and principal properties of the orthogonal trajectories of the parallel s of an elliptical or circular cylinder. Right conoid, having for directrices the axis of the circular cylinder and an orthogonal trajectory. The inter of this conoid by a cylinder about the same axis is an orthogonal trajectory for a series of parallel s.

Third solution: helicoidal. Determination of the angular elevation at which the surfaces of the beds become normal to the head planes; construction in the orthogonal and helicoidal appareil of the curves of junction upon the heads, and the angles which they form with the curves of intrados. Cutting of the stones in these different constructions. Broken helicoidal appareil, for very long skew arches.

Helicoidal trompes at the angles of straight arches; voussures or widenings, which it is necessary to substitute near the heads at the intrados of an arch with a considerable skew; case where the skew is not the same for the two heads. Orthogonal trajectories of the converging s of a cylinder.

Lessons 23–25. Conical Intrados—Intrados of Revolution.

Skew trompe in the angle. Suggestions on the general problem of conical skew vaulted roofs.

Spherical domes, &c.

Lessons 26–27. Intrados, a Ruled Surface.

Winding staircases, &c., &c.

Lesson 28. Helicodial Intrados.

Staircase on the Saint-Giles screw.

Lessons 29–31. Composite Vaulted Roofs.

Various descriptions of vaults.

Suggestions on vaulted roofs with polygonal edges and with ogival edges.

Lesson 32. Revision.

Spirit and method of stereotomy.

Degree of exactness necessary. Approximate solutions. Case where it is proper to employ calculation in aid of graphical constructions.

Review and comparison of different appareils.

MECHANICS AND MACHINES.

GENERAL ARRANGEMENTS.

The pupils execute during the two years of study:—

1. Various drawings or plans of models in relief representing the essential and internal organs of machines, such as articulations of connecting rods, winch-handles and fly-wheels, grease-boxes, eccentrics worked by cams or circles giving motion to rods; the play of slides, &c.; cylinders of steam-engines, condenser, pistons, and various suckers; Archimedes’ screw, and other parts of machines.

The sketches of the plan drawings are traced by hand and figured. The drawings in their finished state are washed and colored according to the table of conventional tints; they all carry a scale suitably divided.

2. A drawing of wheel-work by the method of development, and tracing the curves of teeth by arcs of circles from which they are developed. This drawing represents, of the natural size, or on any other scale of size considered suitable to show the nature of the partial actions only, a small number of teeth either in development or projection; the entire wheel-work is represented by the usual method of projection, where in drawings on a small scale the teeth are replaced by truncated pyramids with a trapezoidal base.

3. Finally, numerical exercises concerning the loss of work due to the proejudicial resistances in various machines, the gauging of holes, orifices, &c.

Models in relief or drawings on a large scale, of the machines or elements of the machines mentioned in the course, assist in explaining the lessons. They are brought back, as often as found necessary, under the eyes of the students. When possible, lithographic sketches of the machines, or the elements of the machines, which ought to enter into the course, are distributed among the pupils.

The pupils, divided into s, pay their first visit to the engine factories towards the end of their first year of study; they make one or more additional visits at the end of the second year.

FIRST YEAR.

PART I. KINEMATICS.—PRELIMINARY ELEMENTARY MOVEMENTS OF INVARIABLE POINTS AND SYSTEMS.

Lessons 1–2.

Object of kinematics, under the geometrical and experimental point of view. Its principal divisions.

Re-statement of the notions relative to the motion of a point, its geometrical representation, and more especially the determination of its velocity.

Simultaneous Velocities of a Point and the Increments of its Velocities.

Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:—Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle.

Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes.

Simple Motions of Solids, or Rigid Systems.

1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points.

2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity.

Geometrical notions and theorems relative to the instantaneous center of rotation of a body of invariable figure and movable in one plane, or to the instantaneous axis of rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples—and amongst others—that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the inter of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point.

COMPOSITION OF MOTIONS.

Lessons 3–6. Composition of the Velocities of a Point.

Polygon of velocities. Example of movements observed relatively to the earth. Particular cases; composition of velocities taken along three axes; composition of the velocity of a point round a fixed pole, and its velocity along the radius vector. Method of Roberval for tracing tangents.

Composition of the Simple Motions of a Solid System.

Composition of any number of translatory displacements of a solid. Composition of two rotations about two intersecting axes. Composition of any number of rotations about axes cutting one another at the same point; parallelopiped and polygon of rotations. Composition of two simultaneous rotations about parallel axes; case where the rotations are equal and of opposite kinds. Decomposition of a rotation about an axis into an equal rotation about any axis whatever parallel to the first, and a translation perpendicular to the direction of this axis. Direct and geometrical decomposition of the most general motions of a body into a rotation about, and a translation along, an axis called the instantaneous axis. Composition of any two motions whatever. Every movement of an invariable system is at each instant of time decomposable into three movements of rotation, and three movements of translation with respect to three axes, which are neither parallel nor lying in the same plane, but otherwise arbitrarily chosen.

Relative or Apparent Motions.

Relative motion of two points whose absolute motions are given graphically à priori. Trajectory of the relative motions, relative velocities, and displacements upon curves or upon the direction of the mutual distance of the two points; use of the parallelogram to determine its amount. Relative motion of a point in motion in respect of a body turning about a fixed axis; relative motion of two bodies which turn about parallel or converging axes, and in general of two rigid bodies or systems impelled by any motions whatever. How this problem is immediately reduced to that of the composition of given motions.

The most general continued motion of an invariable figure in a plane is an epicycloidal motion, in which the instantaneous center describes a curve fixed in relation to absolute space, and traces relatively to the proposed figure a movable curve, which is rigidly connected with that figure and draws it along with it in its motion of rolling upon the other fixed curve. Case of space or spherical figures.

ON THE ACCELERATED MOTION OF A POINT.

Lessons 7–9. Accelerated Rectilinear Motion.

Re-statement of the motions acquired relatively to the acceleration in the variable rectilinear motion of a point. Brief indication of the solution of six problems arising out of the investigation of the laws of the motion in terms of the space, time, velocity, and accelerating force. For the most part these solutions may be brought to depend on exact or approximate quadratures. Numerical exercises.

Accelerated Curvilinear Motions.

Re-statement of the notions acquired relative to the composition of accelerating forces; the resulting acceleration, the normal and tangential acceleration animating a point in motion on a curve. The total acceleration of a point upon an axis or plane is the projection upon this axis or plane of the acceleration of the moving body in space. In uniform curvilinear motion the total or resultant acceleration becomes normal to the curve. Particular case of the circle; value of the normal acceleration in terms of the velocity of revolution or the angular velocity of the radius vector. Case of any curve whatever; geometrical expression of the total or resultant acceleration.

Accelerated Compound and Relative Motions.

Geometrical investigation of the simple and compound accelerations arising out of the hypothesis in which the motion of any system of points whatever is referred to another system of invariable form, but also in motion. Geometrical and elementary explanations of the results obtained by means of the transformation of co-ordinates.

Examples or Exercises chosen from among the following Questions:—

Projection of circular and uniform motion upon a fixed straight line or plane; motion of a circle which rolls uniformly on a straight line; comparison of the motions of the planets relatively to each other, treating them as circular and uniform: comparison of the accelerating force on the moon with that of bodies which fall to the earth.

GEOMETRICAL THEORY AND APPLICATION OF MECHANISMS OR CONTRIVANCES FOR THE TRANSFORMATION OF MOTION.

Lessons 10–19.

Succinct notions on the classification of elementary motions and organs for transmission of motion in machines after Monge and Hachette, Lanz and Bétancourt.

The most essential details upon this subject are set forth in the following order, and made clear by outline drawings previously distributed among the pupils.

Organs fitted to regulate the direction of the circular or rectilinear motion of certain pieces.

Axle; trunnions, gudgeons; pivots and bearings; couplings of axes; adjustment of wheels and of their arms. Joints with hinges, &c.; sheaves and pulleys; chains, ropes, and straps; means of securing them to the necks. Grooves and tongue-pieces. Eyelet-holes sliding along rectilinear or curvilinear rods. Advantages and disadvantages of these different systems of guides under the point of view of accuracy.

Rapid indication of some of their applications to drawbridges and to the movable frames or wagons of saw-works and railways.

Transmission at a Distance of Rectilinear Motion in a determinate Direction and Ratio.

Inclined plane or wedge guiding a vertical rod. Wedge applied to presses. Rods, winch-handles, &c. Disposition of drums or pulleys in the same plane or in different planes; geometrical problem on this subject. Fixed and movable pulleys. Blocks to pulleys. Simple and differential wheel and axle moved by cords. Transmission through a liquid. Ratios of velocities in these different organs.

Direct Transformation of circular progressive motion into progressive and intermittent rectilinear motion.

Rod conducted between guides: 1^o, by the simple contact of a wheel; 2^o, by cross-straps or chains; 3^o, by a projecting cam; 4^o, by means of a helicoidal groove set upon the cylindrical axis of the wheel. To-and-fro movement, and heart-shaped or continuous cam, waves, and eccentrics. Simple screw and nut. Left and right handed screws; differential screw of Prony, called the micrometric screw. Ratio of the velocities in these different organs.

The example of the cam and pile-driver will be particularly insisted upon; 1^o, in the case where this cam and the extremity of the rod have any continuous form given by a simple geometrical drawing; 2^o, in the case where this form is defined geometrically by the condition, that the velocity is to be transmitted in an invariable ratio, as takes place for cams in the form of epicycloids or involutes of circles.

Transformation of a circular progressive motion into another similar to the first.

1^o, by contact of cylinders or cones, the two axes being situated in the same plane; 2^o, by straps, cords, or endless chains, the axes being in the same situation; 3^o, by cams, teeth, and grooves, at very slight intervals; 4^o, by the Dutch or universal joint. Case, where the axes are not situated in the same plane; use of an intermediate axis with beveled wheels or a train of pulleys; idea of White or Hooke’s joint in its improved form. Endless screw specially employed in the case of two axes at right angles to one another. Combinations or groupings of wheels. Idea of differential wheels. Relations of velocities in the most important of these systems of transmission.

Transformation of circular progressive Motion into rectilinear or alternating circular motion.

Ordinary circular eccentric. Eccentrics with closed waves or cams. Examples and graphical exercises in the class-rooms relative to the alternate action of the traveling frames of saw-mills, of the slides or entrance valves of steam-engines. Cams for working hammers and bellows.

Transformation of alternating circular motion into alternating rectilinear motion, or into intermittent and progressive circular motion.

Pump rods with or without circular sectors, &c. Examples taken from large exhausting pumps, fire-engines, and common pumps. Suggestions as to the best arrangement of the parts. Lagarousse’s lever, &c. Application of the principle relative to the instantaneous center of rotation to give the relations of the velocities in certain simple cases.

Transformation of alternating circular or rectilinear motions into progressive circular motion.

The knife-grinder’s treadle. System of great machines worked with connecting rods, fly-wheel, &c. Watt’s parallelogram, and the simplest modifications of it for steamboats, for instance. The most favorable proportions for avoiding the deviation of piston-rods. Simplification of parts in the modern steam-engines of Maudsley, Cavé, &c. Variable ratios of the velocities.

Of organs for effecting a sudden change of motion.

Suspendors or moderators, &c. Dead wheels and pulleys, &c. Mechanisms for stretching cords or straps, and make them change pulleys during the motion. Brakes to windmills, carriages, &c., &c. Case where the axes are rendered movable. Means for changing the directions and velocity of the motions. Coupled and alternate pulleys; alternate cones; castors moving by friction and rotation upon a plate or turning-cone; eccentric and orrery wheels. Means of changing the motion suddenly and by intervals; wheels with a detent pile-drivers; Dobo’s escapement for diminishing the shock, &c.

Geometrical Drawing of Wheel-work.

General condition which the teeth of toothed wheels must satisfy. Consequence resulting from this for the determination of the form of the teeth of one of two wheels, when the form of the teeth of the other wheel is given.

Cylindrical action of toothed wheels or toothed wheels with parallel axes. External engagement of the teeth; internal engagement. Particular systems of toothed wheels; lantern wheels, flange wheels, involutes of circles. Reciprocity of action; case where the action can not be rendered reciprocal. Pothook action. Details as to the form and dimensions given in practice to the teeth and the spaces which separate them.

Conical action of toothed wheels, or toothed wheels with converging axes. Practical approximate method of reducing the construction of a conical to that of a cylindrical engagement of toothed wheels.

Means of observation and apparatus proper for discovering experimentally the law of any given movement.

Simple methods practiced by Galileo and Coulomb in their experiments relative to the inclined plane and the motion of bodies sliding down it. Various means of observing and discovering the law of the translatory and rotatory motion of a body according as the motion is slow or rapid. Determination of the angular velocity, &c. The counter in machines. Apparatus of Mattei and Grobert for assigning the initial velocity of projectiles (musket balls.) Colonel Beaufoy’s pendulum apparatus. Chronometrical apparatus for continuous indications by means of a pencil. Eytelwein’s apparatus with bands, and its simplest modifications. Apparatus with cylinders or revolving disks. Use of the tuning-fork for measuring with precision very small fractions of time.

(The principal sorts of the apparatus above described are made to act under the eyes of the pupils.)

PART II.—EQUILIBRIUM OF FORCES APPLIED TO MATERIAL SYSTEMS.

Lesson 21.

Résumé of the notions acquired upon the subject of forces, and their effects on material points.

Principle of inertia, notion of force, of its direction, of its intensity. Principle of the equality of action and reaction. What is meant by the force of inertia? Principle of the independence and composition of the effects of forces. Forces proportional to the acceleration which they produce on the same body. Composition of forces. Relation between the accelerating force, the pressure, and the mass. Definition of the work done by a force. The work done by the resultant is equal to the sum of the works done by the components. Moment of a force in relation to an axis deduced from the consideration of the work of the force applied to a point turning about a fixed line. The moment of the resultant of several forces applied to a point is equal to the sum of the moments of the components. Corresponding propositions of geometry.

Lessons 22–25.

Succinct Notions upon the Constitution of Solid Bodies.

Every body or system of bodies may be regarded as a combination of material points isolated or at a distance, subject to equal and opposite mutual actions. Interior and exterior forces. Example of two molecules subject to their reciprocal actions alternately, attractive and repulsive, when the forces applied draw them out of their position of natural equilibrium. Different degrees of natural solidity, stability, or elasticity; they can only be appreciated by experience.

Equilibrium of any Systems whatever of Material Points.

General theorem of the virtual work of forces applied to any system whatever of material points. It is applicable to every finite portion of the system, provided regard be had to the actions exercised by the molecules exterior to the part under consideration. Determination of the sum of the virtual works of the equal and reciprocal actions of two material points. Demonstration of the six general equations of equilibrium of any system whatever. They comprise implicitly every equation deduced from a virtual movement compatible with the pre-supposed solidification of the system.

Theorem on the virtual work in the case of systems where one supposes ideal connections, such as the invariability of the distance of certain points of the system from one another, and the condition that certain of them are to remain upon curves either fixed or moving without friction.

Equilibrium of Solid Bodies.

The six general equations of equilibrium are sufficient as conditions of the equilibrium of a solid body. Theory of moments and couples.

APPLICATIONS.

Lessons 26–29. Equilibrium of Heavy Systems.

Recapitulation of some indispensable notions for the experimental determination of the center of gravity of solids when the law of their densities is unknown. Re-statement of the theorem relative to the work done by gravity upon a system of bodies connected or otherwise. In machines supposed without friction submitted, with the exception of their supports, to the action of gravity alone, the positions of stable or unstable equilibrium correspond to the highest or lowest points of the curve which would be described by the center of gravity of the system when made to move. Influence of defect of centering in its wheels, upon the equilibrium of a machine. Case where the center of gravity always remaining at the same height the equilibrium is neutral. Examples relative to the most simple drawbridges, &c.

Equilibrium of Jointed Systems.

Equilibrium of the funicular polygon deduced from direct geometrical considerations: Varignon’s theorem giving the law of the tensions by another polygon whose sides are parallel and proportional to the forces acting upon the vertices of the funicular polygon. Case of suspension bridges; investigation of the curve which defines the boundary of the suspension chain; tensions at the extremities.

Equilibrium of systems of jointed rigid bodies without friction. Determination of the pressure upon the supports and the mutual actions at the joints.

Equilibrium and stability of solid bodies submitted to the action of stretching or compressing forces.

Permanent resistance and limiting resistance of prisms to longitudinal extension and compression. Equilibrium and stability of a heavy solid placed upon a horizontal plane and submitted to the action of forces which tend to overset it. Resultant pressure and mean pressure; hypothetical distribution of the elements of the pressure on the base of support. Conditions of stability, regard being had to the limit of resistance of solid materials, co-efficient of stability deduced from it.

PART III.—ON THE WORK DONE BY FORCES IN MACHINES.

Lessons 30–39. General Notions.

Principle of work in the motion of a material point. Extension of this principle to the case of any material system whatever in motion. Considerations relative to mechanical work in various operations, such as the lifting of weights, sawing, planing, &c. It is the true measure of the productive activity of forces in industrial works. It may always be calculated either rigorously or approximately when the mathematical or experimental law which connects the force with the spaces described is given. Uniform work, periodical work, mean work, for the unit of time. Horse-power unit. Examples and various exercises, such as the calculation of the work corresponding to the elasticity of gases on the hypothesis of Mariotte’s law, the elongation of a metallic prism, &c.

Dynamometrical Apparatus.

Dynamometer of traction by a band or rotating disc or register. Dynamometer of rotation with simple spring, with band or register. Dynamometer of rotation with multiple springs and with register for the axles of powerful machines. Improved indicator of Watt.

(These pieces of apparatus are made to act under the eyes of the pupils.)

Work of Animal Prime Movers upon Machines.

Results of experience as to the values of the daily work which animal motors can supply under different circumstances without exceeding the fatigue which sleep and nourishment are capable of repairing.

Theory of the Transmission of Work in Machines.

Principal resistance. Secondary resistances. Two manners in which bodies perform the duty of motors. Ratio of work done to work expended always inferior to unity. Different parts of machines; receiver; organs of transmission; tools as machines.

Calculation of the Work due to the passive resistances in machines.

Résumé of the notions previously acquired on friction. Application to the inclined plane, to the printing-press, to guides or grooves, to the screw with a square thread; different cases of uniform motion being impossible under the action of forces of given directions. Friction of trunnions, pivots, eccentrics, and insertions of winch-handles. Prony’s dynamometrical brake; conditions of its application. Resistance to rolling; its laws according to experiment. Use of rollers and friction-wheels; their practical inconveniences.

Mixed friction of toothed wheels; the Dobo escapement: friction of the teeth in the endless screw.

Stiffness and friction of cords. Results of experience. Friction of cords and straps running round drums. Different applications; brakes; transmission by cords, endless straps, or chains.

Examples and exercises; effects of passive resistances in the capstan, the crane, pulleys, &c.

Lesson 40. Revision.

SECOND YEAR.

PART I.—DYNAMICS.— DYNAMICS OF A MATERIAL POINT.

Lessons 1–2. Completion of the Notions acquired on this Subject.

Differential equations of the motion of a material point submitted to the continued action of one or more forces. The acceleration of the projection of a point upon any axis or plane is due to the projection of the forces on this axis or plane. The acceleration along the trajectory is due to the tangential force. Relation of the curvature to the centripetal force. Introduction of the force of inertia into the preceding enunciations.

The increase of the quantity of motion projected upon an axis or taken along the trajectory is equal to the impulsion of the projected resultant, or to that of the tangential force. The total impulsion of a force is got by methods of calculation and of experiment analogous to those which relate to work. The increase of the moment of the quantity of motion in relation to any axis is equal to the total moment of the impulsions of the forces during the same interval of time; direct geometrical demonstration of this theorem. In decomposing the velocity of the moving body into a velocity in the plane passing through the axis of the moments, and a velocity of revolution perpendicular to this plane, we may replace the moment of the quantity of motion in space by the quantity of motion of revolution. Particular case known under the name of the principle of areas.

Extension of the preceding theorems to the case of relative motions. Apparent forces which must be combined with the real ones that the relative motion of a point may be assimilated to an absolute motion. Particular case of relative equilibrium. Influence of the motion of the earth upon the accelerating force of gravity.

DYNAMICS OF ANY MATERIAL SYSTEMS.

Lessons 3–8.

Principle or general rule which reduces questions in dynamics to questions in equilibrium by the addition of the forces of inertia to the forces which really act on the system. Equation of virtual work which expresses this equilibrium; it comprises in general the external and internal forces.

General Theorems.

These theorems, four in number, are founded upon the principle of the equality of action and reaction applied to internal forces. They may be deduced from the preceding rule, but the three last are obtained more simply by extending to a system of material points analogous theorems established for isolated material points.

General theorem of the motion of the center of gravity of a system. Particular case called principle of the conservation of the motion of the center of gravity.

General theorem on the quantities of motion and impulsions of exterior forces projected on any axis.

General theorems of moments of quantities of motion and impulsions of exterior forces, projected on any axis whatever.

General theorems of the moments of quantities of motion and impulsions of exterior forces about any axis. Analogy of these two theorems with the equations of the equilibrium of a solid, in which the forces are replaced by impulsions and quantities of motion.

Composition of impulsions, of quantities of motion, or the areas which represent them. All the equations which can be obtained by the application of the two theorems relative to quantities of motion and impulsions, reduce themselves to six distinct equations. Particular case called principle of the conservation of areas. Fixed plane of the resulting moment of the quantities of motion called plane of maximum areas.

General theorem of work and vis viva. Part which appertains to the interior forces in this theorem. Particular case called principle of the conservation of vires vivæ, where the sum of the elements of work done by the exterior and interior forces is the differential of a function of the co-ordinates of different points of the system. Application of the theorem of work to the stability of the equilibrium of heavy systems.

Extension of the preceding theorems to the case of relative motions. Particular case of relative equilibrium. Motion of any material system relative to axes always passing through the center of gravity, and moving parallel to themselves. Invariable plane of Laplace. Relation between the absolute vis viva of a material system, and that which would be due to its motion, referred to the system of movable axes above indicated.

Examples and Applications.

The following examples, amongst others, to be taken as applications or subjects of exercises relative to the general principles which precede.

Walking. Recoil of guns. Eolypile. Flight of rockets.

Pressure of fluid veins, resistance of mediums, &c. Direct collision of bodies more or less hard, elastic, or penetrable. Exchange of quantities of motion. Loss of vis viva under different hypotheses. Influence of vibrations and permanent molecular displacements.

Pile driving; advantage of large rammers. Comparison of effects of the shocks and of simple pressures due to the weight of the construction. Oblique collision, and ricochet. Data furnished by experiment.

Oscillations of a vertical elastic prism suspended to a fixed point, and loaded with a weight, neglecting the inertia, and the weight of the material parts of this prism. Case of a sudden blow. What is meant by the “resistance vive” of a prism to rupture? Results of experiments.

Work developed by powder upon projectiles, estimated according to the vis viva which it impresses on them, as well as upon the gun and the gases upon hypothesis of a mean velocity.

SPECIAL DYNAMICS OF SOLID BODIES.

Lessons 9–12. Simple Rotation of an invariable Solid about its Axis.

In applying to this case the first general rule of dynamics, the theorem of the moments of the quantities of motion, and the theorem of work, we are led to the notion of the moment of inertia; explanation of the origin of this name. The angular acceleration is equal to the sum of the moments of the exterior forces divided by the moment of inertia about the axis of rotation. Sum of the moments of the quantities of motion relative to this axis. Vis viva of a solid simply turning about an axis. What is meant by radius of gyration?

Remind of the geometrical properties of moments of inertia, of the ellipsoid which represents them, of the principal axes at any point, of those which are referred to the center of gravity.

Pressure which a rotating body exercises on its supports. Reduction of the centrifugal and tangential forces of inertia to a force which is the force of inertia of the entire mass accumulated at the center of gravity, and a couple.

Particular case where the forces of inertia have a single resultant; different examples. Center of percussion. Compound pendulum; length of the corresponding simple pendulum. Center of oscillation; reciprocity of the centers or axes of suspension and oscillation. Pressure upon the axis. Influence of the medium; experience proves that the resistance, varying with the velocity, changes the extent of the oscillations, but does not sensibly affect the time. Experimental determination of the center of oscillation and the moment of inertia about an axis.

Motion of an invariable Solid subject to certain Forces.

General notions on this subject. Motion of the center of gravity; motion of rotation about this point.

Lessons 13–19. Various Applications.

Motion of a homogeneous sphere or cylinder rolling upon an inclined plane, taking friction into account.

Motion of a pulley with its axis horizontal, solicited by two weights suspended vertically to a thread or fine string passing round the neck of the pulley, the axle of which rests upon movable wheels. Atwood’s machine serving to demonstrate the laws of the communication of motion.

Motion of a horizontal wheel and axle acted on by a weight suspended vertically to a cord rolled round the axle, or upon a drum with the same axis, and presenting an eccentric mass. To take account of the variable friction of the bearings, and the stiffness of the cord, with recourse, if necessary, to approximation by quadratures. Oscillations of the torsion balance.

Balistic pendulum. Condition that there may be no shock on the axis. Experimental determination of the direction in which the percussion should take place.

Theory of Huyghen’s conical pendulum considered as a regulator of machinery. How to take account of the inertia and friction of the jointed rods, as well as of the force necessary to move the regulating lever, &c.; appreciation of the degree of sensibility of the ball apparatus with a given uniform velocity.

Windlass with fly-wheel. Dynamical properties of the fly-wheel. Reduced formulæ for a crank with single or double action. Advantages and disadvantages of eccentric masses. Tendency of the tangential forces of inertia to break the arms. Numerical examples and computations.

Mutual action of rotating bodies connected by straps or toothed wheels in varying motion.

The wedge and punching-press. Stamping screw or lever used in coining, cams, lifting a pile or a hammer. To take account of the friction during the blow, and afterwards to estimate the loss of vis viva in cases which admit of it.

PART II.—SPECIAL MECHANICS OF FLUIDS.—HYDROSTATICS.

Lessons 20–22.

Principle of the equality of pressure in all directions. Propagation of the pressures from the surface to the interior of a fluid, and upon the sides of the vessel. Equations of equilibrium for any set of forces. Pressure exerted in the containing orifices. Measure of the pressure upon a plain portion of surface inclined or vertical (sluice-gate, embankments, &c.) Center of push or pressure. Pressure against the surfaces of a cylindrical tube. Effect, and resistance to oppose to the pressure. Manometer and piezometer. Equilibrium of a body plunged in a heavy fluid or floating at its surface. Stability of floating bodies. Metacenter. Laws of the pressure in the different atmospheric strata.

HYDRAULICS.

Lessons 23–27. Flow of Fluids through small Orifices.

Study of the phenomena which accompany this flow in the case of a thin envelop and a liquid kept at a constant level. Conditions of this constancy in the level, and the permanence of the motion in general. Motion of the lines of fluid; form; contraction; reversal and discontinuity of liquid veins. Fundamental formulæ for liquids and gases based upon the principle of vis viva, and Bernoulli’s hypothesis of parallel s or Borda’s of contiguous threads. Torricelli’s theorem relative to small orifices. What is called the theoretical expenditure, effective expenditure, and co-efficient of geometrical contraction. Co-efficient deduced from the effective expenditure. Its variations with the volume of the fluid contents, and the form of the inner surfaces of the reservoir. Results of the experiments of Michelotti, Borda, Bossut, &c. Phenomenon of adjutages. Venturi’s experiments; influence of atmospheric pressure; loss of vis viva; reduction of the velocity and augmentation of the expenditure. Results of experience relative to the co-efficient of expenditure, the form and range of the parabolic jets, showing the initial vis viva, and the loss of vis viva.

Large orifices.—Sluice holes and floodgates; reservoirs or open orifices; expenditure; practical formulæ and results of experiment. Influence of the proximity of the sides and the walls. Arrangement to avoid the effects of contraction or the losses of vis viva.

Flow through conducting Pipes and Open Canals.

Practical formulæ relative to the case of uniform s of great length. Measure of the pressures at different points of a conduit-pipe. Expression for the losses of effect due to corners and obstructions. Flow of gases. Principal methods of measuring the volume consumed adopted in practice. Floats. Pitot’s tube. Woltman’s mill. Register mill in air or gas. Waste in such instruments. Modulus and scale for water-supply.

PART III.—DIFFERENT MACHINES CONSIDERED IN THE STATE OF MOTION.

Lesson 28. General Considerations. Résumé of the Notions acquired on this Subject.

Equation of vis viva, and transmission of work in machines, account being taken of the different causes of power and resistance. Physical constitution of machines; receiver, communicators, and operator. Influence of the weights, of frictions, of shocks, and any changes in the vis viva. Parts with continuous or uniform motion, with alternating or oscillating motion. Laws of the motion on starting from rest, and when the stationary condition is established. The positions to which the maximum and minimum of the vis viva correspond are those in which there is equilibrium between all the forces, exclusive of the forces of inertia. Advantage of uniform or periodic motion. General methods for regulating the motion; symmetrical distribution of the masses and strains; flys and various regulators. Brakes and moderators; their inconveniences. Object and real advantages of machines.

Lessons 27–35. Hydraulic Wheels.

Vertical wheels with float-boards, with curved ladles, and with spouts. Figure of the surface of the fluid in these latter. Horizontal wheels working by float-boards, buckets, and reaction. Turbines. Description, play, and useful effects compared according to the results of experiment. Vertical wheels of windmills and steamboats. Screw propeller.

Windmills.

Description. Result of Coulomb’s observations.

On the principal kinds of Pumps.

Special organs of pumps. Valves and pistons, force pump, sucking pump; limit to the rise of the water. Sucking and force pump. Dynamical effects. Indication as to the losses of vis viva and the waste in different pumps. Explanation of the hydraulic ram. Air vessel. Fire pumps. Double action pumps.

Various Hydraulic Machines.

Hydraulic press. Water engine. Exhausting machines; norias; under and overshot wheels; Archimedes’ screw, construction and experimental data.

Lessons 36–39. Steam Engines.

Succinct description of the principal kinds of steam-engine with or without detent. Effects and advantages of the detent. Condenser. Air Pump. Furnace and feeding-pump.

Variable detent. Formulæ and experimental results.

Lessons 40–42. Revision.

Reflections on the totality of the subjects of the course.

IV. PHYSICS.—FIRST YEAR.

GENERAL PROPERTIES OF BODIES.—HYDROSTATICS.—HYDRODYNAMICS.

Lessons 1–5. Preliminary Notions.

Definitions of physics. Phenomena. Physical laws. Experiments are designed to make them spring out of the phenomena. Method of induction. Physical theories; different character of the experimental and mathematical methods.

General Properties of Bodies.

Extension. Measure of lengths. Vernier. Cathetometer. Micrometer screw. Spherometer. Dividing engine.

Divisibility. Porosity. Ideas generally received on the molecular constitution of bodies. These conceptions, which are purely hypothetical, must not be confounded with physical laws. Elasticity. Mobility. Inertia. Forces; their equilibrium, their effects, their numerical estimation.

Weight or Gravity.

Direction of gravity. Plumb-line. Relation between the direction of gravity and the surface of still water.

Weight. Center of gravity.

Experimental study of the motion produced by weight. In vacuum, all bodies fall with the same velocity. Disturbing influence of the air. Inclined plane of Galileo. Atwood’s machine. To prove by experiment; 1^o the law of the spaces described; 2^o the law of velocities. Morin’s self-registering apparatus with revolving cylinder.

Law of the independence of the effect produced by a force upon a body, and the motion anteriorily acquired by this body. Law of the independence of the effects of forces which act simultaneously upon the same body. Experimental demonstration and generalization of these laws. Law of the equality of action and reaction.

Mass. Acceleration. For equal masses the forces are as the accelerations which they produce. Relation between the force, mass, and acceleration. Collision.

General laws of uniformly accelerated motion. Formulæ.

Pendulum. Law of the isochronism of small oscillations and law of the lengths deduced from observation.

Method of coincidences or beats. Use of the pendulum as the measure of time. Simple pendulum; formulæ. Compound pendulum: the laws of the oscillations of a compound pendulum are the same as the laws of the oscillations of a simple pendulum whose length may be calculated.

Determination by means of the pendulum of the acceleration produced by gravity. This acceleration is independent of the nature of the body.

Remark that the formulæ for the motion of oscillation apply to the comparison of forces of any kind, that may be regarded as constant and parallel to themselves in all positions of the oscillating body.

Identity of gravity and universal attraction.

Measure of weights. Balance. Conditions to be attended to in making it. Absolute sensibility; proportional sensibility. Method of double weighing. Details of the precautions necessary in order to obtain an exact weight.

Different States of Bodies. Hydrostatics.

Solids. Cohesion. Transmission of external pressures.

Elasticity. The true laws of elasticity are unknown. Empirical laws in certain simple cases, and for a very small action. Elasticity of compression, extension, torsion. Experimental determination of the co-efficients of elasticity. Limits of elasticity. Limits of tenacity.

Ductility. Temper. Cold hammering. Annealing.

Liquids. Fluidity. Viscosity. Physical laws which form the basis of hydrostatics:—1^o the transmission of external pressures is equal in all directions; 2^o the pressure exercised in the interior of a liquid upon an element of a surface is normal to that element, and independent (as to amount) of its direction. These principles are demonstrated by the experimental verification of the consequences drawn from them.

Application to heavy liquids. Free surface, and surface de niveau. Pressure upon the parts of the containing vessel, and upon the bottom in particular; hydrostatic paradox; verificatory experiments. Haldat’s apparatus. Hydrostatic press.

Application to immersed or floating bodies (principle of Archimedes;) verificatory experiments. (In treating of the equilibrium of floating bodies, the conditions of stability are not gone into.)

Superposed liquids.

Communicating vessels. Water level. Spirit level; its use in instruments.

Densities of solids and liquids. Anemometers.

Compressibility of liquids. Piezometer. Correction due to the compressibility of the solid envelop.

Gas. Expansibility. Other properties common to liquids and gases. Principle of the equal transmission of pressures in all directions. Weight of gases. Pressure due to weight (principle of Archimedes.) Weight of body in air and in vacuum. Aerostation.

Superposed liquids and gases.

Communicating vessels. Barometer.

Detailed construction of barometer. Barometers of Fortin, Gay-Lussac, Bunten. Indication of the corrections necessary.

Mariotte’s law. Regnault’s experiments.

Manometer with atmospheric air—with compressed air. Bourdon’s manometer.

Law of the mixture of gases.

Air pump. Condensing pump.

Primary Notions of Hydrodynamics.

Toricelli’s principle. Mariotte’s vessel and syphon. Uniform flow of liquids. The same of gases.

Molecular Phenomena.

Cohesion of liquids. Adhesion of liquids to solids. Capillary phenomena. Apparent attractions and repulsions of floating bodies.

Adhesion of drops.

Molecular actions intervene as disturbing forces in the phenomena of the equilibrium and motion of liquids.

HEAT.

EFFECTS OF HEAT ON BODIES.

Lessons 6–9. Generalities.

General effects. Arbitrary choice of one of these effects to define the thermometric condition of a body. Conventional adoption of a thermometer. Definition of temperature.

Dilating Effects.

Definition of the co-efficients of linear, superficial, and cubic dilatation. Approximate relation between the numerical values of these three co-efficients. The value of the co-efficient of dilatation depends upon the thermometric substance and the temperature selected as the zero point. It becomes nearly independent of the zero point when the co-efficient is very small.

Relation between volume, density, and temperature. Linear dilatation of solid bodies. Ramsden’s instrument. Cubical dilatation of liquids. Dulong and Petit’s experiments on mercury. Discussion. Regnault’s experiments.

Cubical dilatation of solids and of other liquids when that of mercury is given.

Relations between the volume, density, and elasticity of a gas, and its temperature.

Cubical dilatation of gases. Experiments of Gay-Lussac, Rudberg, and M. Regnault. Advantage of varying the methods of experimenting in these delicate researches.

Methods based upon the changes of volume under a constant pressure, and upon the changes of pressure for a constant volume.

The disagreement of these two methods is due to deviations from the law of Mariotte.

The constancy of the co-efficients of dilatation previously defined is only approximately true.

Necessity of employing two different co-efficients of dilatation according as consideration is being had to the variations of volume to a given pressure, or of pressure to a given volume.

Empirical formulæ for the dilatation of liquids.

Graphical constructions.

Lesson 10. Thermometers.

Construction of thermometers. Mercurial thermometer. Details of construction. Fixed points. Different scales; their relation. Arbitrary scales. Change which takes place in the zero point. Different precautions to be observed in using the mercurial thermometer.

General want of comparability of mercurial thermometers with tubes of different material.

Air thermometers. They are comparable with one another within the limits of the errors of experiment, whatever the nature of the tube employed. This property entitles the air thermometer to a preference for all accurate measures. Comparison of the air and mercurial thermometers.

THERMOSCOPE, DIFFERENTIAL THERMOMETER, PYROMETERS, BREGUET’S THERMOMETER.

Lessons 11–13. Changes of State produced by Heat.

Exposition of the phenomena which accompany the liquefaction of solids and the solidification of liquids. Constancy of the temperature whilst the phenomenon is going on.

Sudden melting and freezing. Persistance of the liquid state beneath the melting point.

Influence of pressure.

Exposition of the phenomena which accompany the conversion of liquids or solids into vapor, and the inverse passage from the gaseous to the liquid or solid state. Constancy of the temperatures whilst the phenomenon is going on.

Influence of pressure.

Phenomena of ebullition in free space. Augmentation of the temperature and pressure in a confined space. Papin’s digester.

Properties of vapors in spaces and in gases. Saturated vapors. Their tension does not depend upon the space which they occupy, but only upon their temperature.

Effects of a diminution or increase of pressure without change of temperature; the same without change of pressure. Effects of lowering the temperature in a limited region of space occupied by vapor.

Tension of a saturated vapor at the boiling point of its liquid.

Measure of the tensions of the vapor of water. Experiments of Dalton, Gay-Lussac, Dulong, and Arago, and of M. Regnault.

Tables of the tensions of steam. Empirical formulæ. Graphical constructions.

It is assumed that non-saturated vapors are subject to the same laws as gases.

APPLICATIONS. CORRECTION OF THE BOILING-POINT IN THE CONSTRUCTION OF THERMOMETERS. BAROMETRICAL THERMOMETERS.

Lessons 14–16. Various Applications of the Laws previously established.

A phenomenon can not always be separated from the accessory phenomena which concur with it in producing the final result. Necessity of corrections to render complex results comparable inter se.

Density of solids when regard is had to the temperature and weight of the gases displaced by them.

Precautions to be attended to in the experiments. Empirical formulæ for the density of liquids. Maximum density of water. The temperature corresponding to the maximum must be determined graphically, or by interpolation.

Corrections for measures of capacity, for barometric measures.

The uncertainty of the corrections can not, in any considerable degree, affect the densities of solids and liquids.

Density of gases. Biot and Arago’s experiments. Special difficulties of the question. The uncertainty of the corrections may sensibly affect the results. Regnault’s method.

The same method may be applied to the determination of the co-efficient of dilatation for gases.

Density of vapors. Definition founded on the hypothetical application of the same laws to gases and vapors. Formulæ. Experimental method of Gay-Lussac and of Dumas. Corrections. Comparison of the two methods. Necessity of conducting the experiments at a distance from the saturation point. Latour’s experiments. Relations between the weight and volume of a gas, and its temperatures; between the weight and volume of a gas mixed with vapors, and its temperature. Various problems.

Hygrometry. Chemical hygrometry. Hygrometry by the dew-point. Psychrometry.

PROPAGATION OF HEAT.

Lessons 17–18. Propagation at a Distance.

Rapid propagation of heat at a distance, in vacuum, in gases, in certain liquid or solid mediums. Experiments which establish this.

Rays of heat. Velocity of propagation. Intensity of heat received at a distance. Intensity of heat received or emitted obliquely. Emitting power, power of absorption, reflection, diffusion. The emitting and absorbing power are expressible by the same number in terms of their proper units respectively.

Analysis of calorific radiations by absorption. Different effects of deathermanous or thermochroic medium. Different influences of increasing thicknesses of the combination of different mediums. Radiations proceeding from different sources, various effects of different mediums on these radiations.

The calorific radiations emanating from different sources, have all the characters of differently colored heterogeneous rays of light.

THEORY OF RADIATION AND OF THE DYNAMICAL EQUILIBRIUM OF TEMPERATURES. APPARENT REFLECTION OF COLD.

Lesson 19. Law of Cooling.

Definition of the rate of cooling. Many causes may conspire in the cooling of a body.

Cooling in space. Newton’s law only an approximation. Experimental investigation of the true law. Method to be followed in this investigation. The velocity of cooling is not a datum directly observable. It must be deduced provisionally from an empirical relation between the temperature and the time. Preliminary experiments. Course of the definitive experiments. Elementary experimental laws.

Hypothetical form of the function which expresses the velocity of cooling. To determine by means of the preceding experimental laws the unknown form of the function which expresses the law of radiation. Relation between the temperatures and the times. This relation only contains data immediately observable, and may be verified à posteriori.

The contents which enter into the preceding relation depend upon thermometric constants and the nature of the radiating surface.

The contact of a gas modifies the law of cooling.

Lessons 20–21. Propagation by Contact.

Slow propagation of heat in the interior of bodies, in solids, liquids, and gases. Confirmatory experiments. Hypothesis of partial radiation. Theoretical law resulting from this hypothesis upon the decrease of temperatures in a solid limited by two indefinite parallel planes maintained at constant temperatures. Determination of the co-efficient of conductibility by the experimental realization of these conditions. This experiment determines a numerical value of the co-efficients; it is not of a nature to serve as a check upon the theoretical principles. Enunciation of the law resulting from the same theoretical principles upon the decrease of temperatures in a thin bar heated at one end.

CALORIMETRY.

Lessons 22–23. Specific Heats.

Comparison of the quantities of heat. The quantities of heat are not proportioned to the temperatures. Definitions of the unity of heat. General method of mixtures to estimate the quantities of heat. Experimental precautions and corrections.

Application of the general method of mixtures. Specific heats of solids and liquids. Law of the specific heat of atoms. Heat absorbed by expansion, restored by the compression of bodies. Experiments on gases. Specific heats of gases under constant pressure. Measure of specific heats of gases under constant pressure. Special difficulties of the question. Succinct indication of one of the methods. Specific heats to a constant volume.

Lesson 24. Latent Heat.

Component heat of liquids absorbed into the latent state during fusion, restored to the free state during solidification.

Influence of the viscous state. Latent heat of ice. Ice calorimeter; its defects.

Component heat of vapors, absorbed into the latent state during vaporization, restored to the free state during condensation. Measure of the latent heat of vapors. Regnault’s experiments.

Empirical laws on the latent heat of vaporization.

Applications of Calorimetry.

Means of producing heat or cold; 1, by changes in density; 2, by changes of state. Freezing mixtures. Vaporization of liquids. Condensation of vapors.

Steam-boilers. Warming by hot air and hot water. Various problems. Sensations produced by a jet of vapor.

Different physical and chemical sources of heat; percussion, friction, chemical combinations, animal heat, natural heat of the globe, solar heat, &c. It will be remarked that mechanical work may become a source of heat, and heat a source of mechanical work.

STATICAL ELECTRICITY.—MAGNETISM.—STATICAL ELECTRICITY.

Lessons 25–27.

General phenomena. Distinction of bodies into conductors and non-conductors. Distinction of electricity into two kinds. Separation of the two electricities by friction. Hypothesis of electric fluids. Effects of vacuum of gases and vapors of points. Electrical attractions and repulsions. Electrization by influence. Case where the influenced body is already electrized. Sparks; power of points. Electrization by influence preceding the motion of light bodies.

Electroscopes.

Electrical machines of Van-Marum, Nairne, Armstrong.

Condenser. Accumulation of electricity upon its surface. Leyden jar. Batteries. Electrical discharges. Effects of electricity.

Condensing electroscope. Electrophorus.

Velocity of statical electricity.

Atmospherical electricity. Phenomena observed with a serene sky. Electricity of clouds. Storms. Lightning. Thunder. Effects of thunder. Return-shock. Lightning conductor.

Different sources of statical electricity.

MAGNETISM.

Lessons 28–30.

Natural magnets. Action upon iron and steel. Artificial magnets. The attractive action appears as if it were concentrated about the extremities of magnetic bars. First idea of poles.

Direction of a magnetized bar under the earth’s action. Reciprocal action of the poles of two magnets. Names given to the poles.

Phenomena of influence. Action of a magnet upon a bar of soft iron; upon a bar of steel. Coercive force. Effects of the rupture of a magnetized bar. Theoretical ideas on the constitution of magnets. More precise definition of the poles.

Action of the earth upon a magnet. The earth may be considered as a magnet. Its action may be destroyed by means of a magnet suitably placed. Astatic needles. The magnetic action of the earth is equivalent to a couple. Three constants define the couple of terrestrial action. Declination. Inclination. Intensity. Measure of the declination; of the inclination.

Magnetic metals. Influence of hammering, tempering, &c. Methods of magnetizing. Saturation. Loss of magnetism. Influence of heat. Magnetic lines. Armatures.

Magnetization by the earth’s influence. Means of determining the magnetic state of a body.

Measure of Magnetism and Electricity.

Lessons 31–32.

Coulomb’s balance. Distribution of magnetism on a magnetized bar; distribution of electricity at the surface of isolated conductors. Comparative discussion of the conditions of the two problems and the methods of experiment.

Laws of the magnetic attractions and repulsions. Law of electric attractions and repulsions. Comparative discussion of the conditions of the two problems, and the methods of experiment.

Determination of the law of magnetic attractions and repulsions by the method of oscillations.

Comparison of the magnetic intensity at different points of the earth’s surface.

Lessons 33–34. Revision.

Considerations on the totality of the subjects of the course.

SECOND YEAR.

DYNAMICAL ELECTRICITY.—GALVANISM.

Lessons 1–2.

Chemical sources of electricity. Experimental proofs. Arrangement devised by Volta to accumulate, at least in part, at the extremities of a heterogeneous conductor the electricity developed by chemical actions.

Pile. Tension at the two isolated extremities; at one single isolated extremity; at the two extremities reunited by a conductor. Continuous current of electricity. Poles. Direction of the current, &c.

Various modifications of the pile of Volta. Woollaston’s pile, Münch’s pile, &c. Dry piles; their application to the electroscope.

Principal effects of electricity in motion, and means of making the currents perceptible. Experiment of Oersted. Galvanoscopes.

Currents produced by heat in heterogeneous circuits. Thermo-electric piles. Thermometric graduation of thermo-electric piles.

Currents produced by the sources of statical electricity.

PROPERTIES OF CURRENTS.

Lesson 3. 1. Chemical Actions.

Definitions. Phenomena of decomposition and transference. Reaction of the elements transferred upon electrodes of different kinds.

Principles of electrotyping.

Causes of the variation of the current in ordinary piles; means of remedying this; Daniell’s pile. Bunsen’s pile.

Lessons 4–8. 2. Mechanical Properties.

Reciprocal actions of rectilinear or sinuous currents parallel or inclined. Reaction of a current on itself.

Reciprocal actions of helices or solenoids. Continuous rotation of currents by their mutual action; by reaction. Analogy of magnets and solenoids. Electro-dynamical theory of magnetism. Action of magnets upon currents and solenoids. Action of currents upon magnets. Experiments of Biot and Savart. Continual rotation of a current by a magnet; of a magnet by a magnet.

Action of the earth upon currents; it acts as a rectilinear current directed from east to west, perpendicularly to the magnetic meridian.

Continual rotation of a current by the action of the earth.

Astatic conductors.

Lessons 9–10. 3. Magnetic Properties.

Action of an interposed conductor upon iron filings.

Electro-magnets. Magnetization temporary or permanent. Principles of the electric telegraph. Electrometers. Reference to diamagnetic phenomena.

4. Electro-motive Properties.

Phenomena of induction by currents, by magnets. Phenomena of magnetism in motion. Induction of a current upon itself.

Induction of different orders.

Interrupted currents. Clarke’s machine.

Lesson 11. 5. Calorific Properties.

Influence of the nature of the interposed conductor; of its ; of the intensity of the current. Unequal temperatures at the different junctions of a heterogeneous circuit.

6. Luminous Properties.

Incandescence of solid conductors. Spectrum of the electric light. Voltaic arc. Transfer of ponderable matter. Action of the magnet upon the Voltaic arc.

7. Physiological Action of Currents.

Some words on this subject. Muscles and nerves. Actions of discontinuous currents. Reotomic contrivances.

Reometry.

Compass of sines, of tangents. Experimental graduation of galvanometers.

The dynamical intensity of a current diminishes when the length of a current increases. Reostat.

Laws of the dynamical intensity of a current in a homogeneous circuit. Reduced length and resistance of a circuit. Specific co-efficients of resistance. Laws of the dynamic intensity of a current in a heterogeneous circuit.

The intensity of currents is in the inverse ratio of the total reduced length, and proportional to the sum of the electromotive forces. Formula of the pile. Discussion of the case of hydro-electric piles—thermo-electric piles. Conditions for the construction of a pile, with reference to the effects to be produced. Conditions for the construction of a galvanometer with reference to its intended application.

Laws of secondary currents in the simplest cases. The chemical intensity of a current is proportional to its dynamical intensity.

ACOUSTICS.

Lessons 12–15.

Noise, sound, quality of the sound, pitch, intensity, timbre. A state of vibration in a solid, liquid, or gaseous body is accompanied with the production of sound.

The pitch depends on the number of vibrations. Unison. Instruments for counting the vibrations:—1st. Graphic method. 2nd. Toothed wheels. 3rd. Lever. Feeling of concord. Musical scale. Gamut. Limit of appreciable sounds.

Study of vibrating motions in solids. Vibrating cords. Vibrations transversal, longitudinal. Experimental laws. Sonometer.

Spontaneous division of a cord into segments. Fundamental sounds. Harmonic sounds.

Straight and curved rods. Transversal and longitudinal vibrations. Experimental laws. Division into segments. Nodes. Ventral segments. Membranes.

Plane and curved plates. The vibrations divide them into “concamerations.” Nodal lines. Harmonic sounds.

Study of the vibrations in liquids and in gases.

Theoretical ideas upon the propagation of a vibratory motion in indefinite elastic media, on an indefinite cylindrical tube. Waves of condensation of dilatation. Progressive nodes and ventral divisions. Laws of the intensities of sound. Direct measure of the velocity of the propagation of sound in water. Measure of the velocity of the propagation of sound in air. Formulæ without demonstration. Comparison of the formulæ with experiment.

Sonorous waves reflected in an indefinite medium.

Fixed nodes and ventral divisions. Sonorous waves reflected in closed and open tubes. Fixed nodes and ventral divisions; the vibratory state and density thereat.

Series of sounds afforded by the same tube. Effect of holes.

Sonorous reflected waves in rods. Series of sounds afforded by the same rod vibrating longitudinally. Indirect measure of the velocity of sound in gases, liquids, and solids.

Experiments on the communication of vibrating motion in heterogeneous mediums, on the general direction of the vibrating motion communicated.

Intensification of sounds. Interferences. Beats. Different stringed and wind instruments. Means of setting them in vibration.

A few words on the organs of voice and hearing. Incompleteness of our knowledge on this subject.

OPTICS.

Lessons 16–17. Propagation of Light.

Propagation of light in a straight line. Rays of light. Geometrical theory of shadows. Velocity of light. Rœmer’s observations. Laws of intensity of light. Photometers of Bouguer, Rumford. Intensity of oblique rays. Comparison of illuminating powers. Total brightness. Intrinsic brightness.

Reflection.

Reflection of light: its laws. Experimental demonstration. Images formed by one or more plane mirrors. To ascertain if a looking-glass has its two faces parallel.

Spherical mirrors. Foci, formulæ. Discussion. Images by reflection. Measure of the radius of a spherical mirror.

Definition of caustics by reflection. Definition of the two spherical aberrations in mirrors.

Woollaston’s goniometer.

Lesson 18. Refraction.

Refraction of light in homogeneous mediums. Descartes’ law. Experimental demonstration for solids and liquids.

Inverse return of the rays. Successive refractions. Indices of transmission in terms of the principal indices. Consequences of Descartes’ law. Total reflection. Manner of observing it.

Irregular refractions. Mirage.

Refraction is always accompanied with the accessory phenomenon of dispersion.

Geometrical consequences of the law of refraction. Focus of a plane surface. Focus of a medium bounded by two parallel plane surfaces; by two plane surfaces inclined in the form of a prism.

Foci of a spherical surface; of a medium limited by two spherical surfaces. Lenses.

Formula for lenses. Discussion. Varieties of lenses. Optic center. Images. Measure of the focal distance of lenses.

Definition of caustics by refraction. Definition of the two spherical aberrations of a lens.

Lessons 19–20. Dispersion.

Unequal refrangibility of the differently colored rays which compose white light. Analysis of heterogeneous light by the prisms. Newton’s method. Solar spectrum. Homogeneity of the different colors. Second refraction of a homogeneous pencil. Experiment with crossed prisms. Precautions to be attended to in the experiments. The spectrum, obtained by Newton’s method, differs from the spectrum produced at the focus of a lens placed between the prism and the picture, according to the method of Fraunhofer. Reasons of the comparative purity of this latter spectrum. Fraunhofer’s lines. Different spectra of different sources of heterogeneous light. Marginal iridescence of a large pencil of natural light traversing a prism. Dispersion of light by lenses. Iridescence of focal images. Recomposition of light, by means of a prism at the focus of a spherical mirror or a lens, by the rapid rotation of a plane mirror, by the rotation of a disk with party-colored sectors. Compound colors.